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Journal of Computational Physics 407 (2020) 109253

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Journal of Computational Physics

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Arbitrary Lagrangian–Eulerian finite element method for curved and deforming surfacesI. General theory and application to fluid interfaces

Amaresh Sahu a,∗, Yannick A.D. Omar a, Roger A. Sauer b, Kranthi K. Mandadapu a,c

a Department of Chemical & Biomolecular Engineering, University of California, Berkeley, CA 94720, USAb Aachen Institute for Advanced Study in Computational Engineering Sciences (AICES), RWTH Aachen University, Templergraben 55, 52056 Aachen, Germanyc Chemical Sciences Division, Lawrence Berkeley National Laboratory, CA 94720, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 24 February 2019Received in revised form 7 January 2020Accepted 8 January 2020Available online 13 January 2020

Keywords:Interfacial flowsFluid filmArbitrary Lagrangian–EulerianFinite element methodDifferential geometryLipid membrane

An arbitrary Lagrangian–Eulerian (ALE) finite element method for arbitrarily curved and deforming two-dimensional materials and interfaces is presented here. An ALE theory is developed by endowing the surface with a mesh whose in-plane velocity need not depend on the in-plane material velocity, and can be specified arbitrarily. A finite element implementation of the theory is formulated and applied to curved and deforming surfaces with in-plane incompressible flows. Numerical inf–sup instabilities associated with in-plane incompressibility are removed by locally projecting the surface tension onto a discontinuous space of piecewise linear functions. The general isoparametric finite element method, based on an arbitrary surface parametrization with curvilinear coordinates, is tested and validated against several numerical benchmarks. A new physical insight is obtained by applying the ALE developments to cylindrical fluid films, which are computationally and analytically found to be stable to non-axisymmetric perturbations, and unstable with respect to long-wavelength axisymmetric perturbations when their length exceeds their circumference. A Lagrangian scheme is attained as a special case of the ALE formulation. Though unable to model fluid films with sustained shear flows, the Lagrangian scheme is validated by reproducing the cylindrical instability. However, relative to the ALE results, the Lagrangian simulations are found to have spatially unresolved regions with few nodes, and thus larger errors.

© 2020 Elsevier Inc. All rights reserved.

1. Introduction

In this paper, and the subsequent manuscript in the series1 [1], we develop an arbitrary Lagrangian–Eulerian (ALE) theory for arbitrarily curved and deforming two-dimensional interfaces with in-plane fluidity. The theory is based on a surface discretization which is independent of the in-plane material flow, such that the surface mesh need not convect with the material. Consequently, two-dimensional materials with large in-plane flows on arbitrarily deforming surfaces can be

* Corresponding author.E-mail addresses: [email protected] (A. Sahu), [email protected] (Y.A.D. Omar), [email protected] (R.A. Sauer),

[email protected] (K.K. Mandadapu).1 From now on, we refer to the present paper as “Part I” and the following one [1] as “Part II.”

https://doi.org/10.1016/j.jcp.2020.1092530021-9991/© 2020 Elsevier Inc. All rights reserved.

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2 A. Sahu et al. / Journal of Computational Physics 407 (2020) 109253

modeled. In Part I, we develop our theory and use standard numerical techniques to devise an isoparametric ALE finite element method for incompressible fluid films. We then implement the finite element formulation, model the deformations and flows of such materials over time, and provide several numerical results for both flat and cylindrical geometries. In Part II, we hope to extend the finite element formulation to lipid membranes and study membrane behavior in several biologically relevant situations. As the equations governing single- and multi-component lipid membranes reduce to the fluid film equations in the limit where no elastic energy is stored in the membrane, such a separation is natural and allows us to present our results in a more accessible manner.

Two-dimensional fluids have played an increasingly important role in many engineering applications, in which they often arise at phase boundaries in multiphase systems [2]. For example, under the influence of gravity and capillary forces, foams will drain over time until the constituent bubbles burst [3]. Foam lifetime plays a key role in their viability for engineering applications, and there have consequently been many efforts to improve foam stability [2]. Similar efforts have been made to stabilize emulsions and colloidal dispersions, which again are of much industrial value [4,5]. Surfactants are often used to stabilize vapor–liquid and liquid–liquid interfaces by lowering the local surface tension [6]; surface tension gradients can drive Marangoni flows [7–9] and in some cases have been shown to significantly affect material properties [10].

Two-dimensional materials with in-plane fluidity also play a fundamental role in biology. Biological membranes, which are interfaces composed of lipids and proteins, are in-plane fluid and out-of-plane elastic materials [11]. They make up the boundary of the cell as well as many of its internal organelles, including the nucleus, endoplasmic reticulum, and Golgi complex. Lipid membranes thus add structure and organization to the cell, and furthermore play an important role in many cellular processes. Endocytosis, for example, begins when proteins in the surrounding bulk fluid bind to the cell membrane’s constituent lipids and proteins at a specific location. The membrane forms an initially shallow invagination, which then develops into a mature bud and eventually pinches off into a membrane-bound vesicle that enters the cell [12]. The vesicular membrane contains lipids and proteins which were previously on the cell boundary, and furthermore the vesicle may enclose nutrients or other cargo. Endocytosis is thus a key process in transferring nutrients to the cell, regulating the expression of proteins on the cell surface, and cell homeostasis [13]. It involves nontrivial coupling between protein binding and unbinding reactions, in-plane lipid flow, and out-of-plane membrane shape changes. In particular, the in-plane flow and out-of-plane bending are coupled because lipid membranes are nearly area-incompressible [11] and therefore lipids are required to flow in-plane to accommodate any shape changes. In another example, lipid membranes can phase separate into liquid–ordered (Lo) and liquid–disordered (Ld) domains under physiological conditions [14]; the energetic penalty of the Lo–Ld interface plays a major role in the fusion of HIV-containing vesicles with target immune cells [15]. This phenomenon demonstrates the value in understanding the coupling between elastic membrane shape changes and the thermodynamically irreversible processes of in-plane lipid flow and in-plane species diffusion.

The interfacial materials discussed thus far are of fundamental importance to engineering and biology. Consequently, there have been significant theoretical efforts to better understand their physics. The pioneering work of L.E. Scriven [16] and R. Aris [17] was crucial to our current understanding of interfacial flows. In particular, Scriven recognized it is prohibitively difficult to use standard Cartesian, cylindrical, and spherical coordinate systems to solve for fluid flows on arbitrarily curved surfaces, where even expressing the surface Laplacian of the velocity field at every point on the surface is nontrivial. Con-sequently, Scriven used a mathematically elegant differential geometric framework to naturally represent two-dimensional flows and their gradients on arbitrarily curved surfaces [16]. Subsequently, Aris worked to describe three-dimensional fluids using the machinery of differential geometry, and incorporated Scriven’s surface flow description within his general differ-ential geometric perspective [17]. The powerful formalism developed by Scriven and Aris continue to be in widespread use today. An excellent review of the interfacial dynamics of fluid interfaces in multiphase systems is provided in Ref. [6], and for a wonderful perspective on interfaces in fluid mechanics see Ref. [18].

While the equations of motion characterizing two-dimensional interfacial flows on surfaces are now widespread and well-understood, theoretical developments for lipid membranes are in a less mature stage. A major complexity arises in modeling lipid membranes because they behave as in-plane fluids, out-of-plane elastic solids, and the surface on which dynamical equations are to be written is itself curved and deforming over time. Early membrane models were modifications of P.M. Naghdi’s seminal contributions to shell theory. However, while Naghdi used a balance law formulation [19], the first membrane models used variational methods and focused only on elastic membrane behavior. In particular, P. Canham [20] and W. Helfrich [21] proposed an elastic membrane bending energy in the early 1970’s; Helfrich also used variational methods to determine the Euler–Lagrange equations governing axisymmetric membrane shapes in the absence of in-plane flows. The Euler–Lagrange equations, which by construction include only thermodynamically reversible phenomena and thus do not contain viscous forces, were not extended to non-axisymmetric settings until 1999 [22]. However, by this time various other models which restricted membrane shapes to small deviations from flat planes [23–26] and cylindrical or spherical shells [26,27] had also emerged. Since then, variational methods encompassing different physical phenomena have continued to be developed [28–36]. In a parallel development, in-plane velocities were included in some models about simple geometries [26,34,37]. It was not until 2009, however, that the general equations governing a single-component, arbitrarily curved and deforming lipid membrane with in-plane viscosity were determined [38]—using a combination of variational methods to determine elastic contributions and the so-called Rayleigh dissipation potential to determine the viscous terms. Since then, variational methods have been extended, with viscous stresses sometimes included in an ad-hoc manner [39–41]. Membrane models have also recently been developed by building on the work of Naghdi [19] and using fundamental balance laws and associated constitutive equations [42–45].

A. Sahu et al. / Journal of Computational Physics 407 (2020) 109253 3

While such theoretical developments have had success in modeling certain membrane phenomena, they were difficult to extend to study how elastic out-of-plane membrane bending is coupled to different irreversible phenomena, such as in-plane lipid flow, in-plane phase transitions involving multiple components, and chemical reactions between membrane components and species in the surrounding bulk. Our recent work [46], inspired by the pioneering works of I. Prigogine [47], L. Onsager [48,49], and S.R. de Groot & P. Mazur [50], developed the general theory of irreversible thermodynamics for arbitrarily curved lipid membranes, provided a formalism to determine the equations governing membrane dynamics, and presented comprehensive models for all of the irreversible phenomena described thus far.

Though the equations governing both fluid interfaces and lipid membranes are now determined, the equations are highly nonlinear and in general cannot be solved analytically. Our work entails developing an ALE theory for two-dimensional materials with in-plane fluidity. The theory involves a surface discretization whose in-plane velocity can be (i) zero, as in an Eulerian formulation, (ii) equal to the in-plane material velocity, as in a Lagrangian formulation, or (iii) specified arbitrarily. The flexibility of our ALE theory, as well as its similarities to bulk methods of the same name [51], explain our nomenclature. With the ALE theory, we numerically solve the equations of motion governing the aforementioned materials of interest. We split this effort into two pieces: in Part I we derive the general ALE theory, develop its finite element formulation, and apply it to two-dimensional fluid films. In Part II we extend the finite element formulation to lipid membranes, which elastically resist out-of-plane bending, and present results from our numerical simulations.

The challenges in theoretically modeling deforming interfaces with in-plane flow, and lipid membranes especially, extend to their numerical modeling as well: to model a material with arbitrarily large shape deformations and in-plane flows, standard techniques from fluid mechanics and solid mechanics are insufficient. Regarding fluid films, many previous studies simplified the problem by assuming the film was fixed in space. The resultant fixed-surface flow equations, derived by Scriven [16], have been solved using various methods: by modeling the fluid interface as a level set in R3 [52], using a vorticity–stream function formulation [53], with surface finite element methods [54,55], and through the discretization of exterior calculus operators [56]. On the other hand, a different study using level set methods [57] made considerable advances in numerically modeling bubble deformation and breakup in foams, however they separated the dynamics into different steps and in each step made simplifying assumptions. In addition, interfaces have been modeled as the boundary between bulk fluid domains, using both ALE and level set techniques—however, such works are either restricted to simple geometries [58,59] or do not include in-plane interfacial flow [60–63]. ALE methods were also used to study the evolution of scalar fields on surfaces whose time evolution is known [64]. While the numerical methods discussed thus far have modeled different fluid film phenomena, they do not seem easily amenable to the study of general deforming fluid interfaces or lipid membranes, with the latter having their own constitutive behavior.

Just as in the case of fluid films, several lipid membrane studies assumed the membrane shape to be fixed, and under these conditions studied how surface flows are coupled to flows in the surrounding bulk in two cases: spherical surfaces with protein inclusions [65] and radial surfaces in a one-to-one correspondence with a sphere [66]. Alternatively, many of the studies modeling the deformation of lipid membranes [67–75] consider only the Euler–Lagrange equations, and thus predict membrane shapes without knowledge of the in-plane flow. However, as the in-plane flow and out-of-plane deformations are coupled through the in-plane viscosity [46], the predictions of the previously mentioned works are only physically relevant in the limit where velocities are negligible. Another approach has been to include in-plane fluid flow and limit the membrane to remain in one-to-one correspondence with a flat plane [42,44]. While such an approach is theoretically sound, it is limited in its use as it cannot, for example, model the large shape deformations observed in endocytosis [12]. Several recent works have avoided the computational complexity of modeling the full lipid membrane equations by assuming only axisymmetric shapes [45,76], however this turns out to be a poor assumption which in many cases yields incorrect results [77]. In our previous work we modeled the full non-axisymmetric membrane equations using a Lagrangian finite element method [77,78], which is computationally valid yet can attain locally singular Jacobians and uninvertible matrices when there are moderate in-plane flows. Lagrangian methods are thus not suitable for the study of general fluid and lipid membrane phenomena.

The limitations of our Lagrangian finite element formulation and other computational techniques in modeling fluid inter-faces and lipid membranes motivate our development of an ALE finite element method for curved and deforming surfaces. The following aspects are new in this work: we

1. develop an ALE theory, within a differential geometric setting, for general arbitrarily curved and deforming two-dimensional interfaces with in-plane flow,

2. apply the theory to two-dimensional fluid films and derive a corresponding isoparametric, fully implicit ALE finite ele-ment method,

3. prevent numerical inf–sup instabilities associated with the in-plane areal incompressibility by adapting the method of C.R. Dohrmann and P.B. Bochev [79] to curved surfaces,

4. numerically simulate an arbitrarily curved and deforming fluid film, from which we find a physical instability that is confirmed analytically with a linear stability analysis, and

5. demonstrate how our ALE formulation can be altered to yield a Lagrangian scheme as a special case.

As mentioned earlier, we limit our numerical calculations to fluid films in this manuscript, as the extensions of the theory and numerical methods to lipid membranes will be presented in Part II [1]. We note Ref. [80] describes a concurrent effort


Fig. 1. A schematic of how the convected (a), surface-fixed (b), and ALE (c) surface coordinates evolve in time, for a side view of the surface. In all cases, the initial surface is represented with a dotted line, the final surface is represented by a solid line, and the circles represent nodes of constant coordinate values. (a) The convected coordinates ξα move along the direction of the material velocity (red arrows), and can squeeze together or spread apart. Moreover, when there are in-plane flows, elements can become highly distorted—leading to complications in numerical implementations, and singularities in extreme cases [77,78]. (b) The surface-fixed coordinates θα only move orthogonally to the surface (solid green arrows), and thus maintain regularity for arbitrarily large in-plane flows. However, as shown in the center of (b), in certain cases it is possible for the surface-fixed coordinates to squeeze together when the surface deforms. (c) The motion of the ALE coordinates ζα can be specified arbitrarily, provided the normal components of the material velocities (transparent red arrows) and mesh velocities (solid blue arrows) are equal, according to Eqs. (7) and (8)—or equivalently Eq. (12). Thus, a regular mesh can in principle always be maintained. (For interpretation of the colors in the figure, the reader is referred to the web version of this article.)

with a similar objective. In particular, Ref. [80] also derives a general ALE theory. However, their numerical implementation is based on a surface parametrization involving small membrane deformations, generalized to arbitrarily curved surfaces, with periodic updates of the reference surface as required. Additionally, Ref. [80] uses a Hodge decomposition of the membrane velocity field, while the present work employs isoparametric finite element methods.

Our paper is organized as follows: In Sec. 2, we present our ALE theory for general two-dimensional interfaces. We pro-vide the equations of motion governing arbitrarily curved and deforming fluid films in Sec. 3 and develop the corresponding finite element formulation in Sec. 4. Numerical results of an ALE implementation are presented in Sec. 5; the modifications leading to a pure Lagrangian formulation and corresponding results are provided in Sec. 6. We end with conclusions and avenues for future work in Sec. 7. Several of the more detailed calculations regarding fluid films and our finite element im-plementation, as well as additional numerical benchmarks, are relegated to Appendices A–D. Relevant movies are provided in Appendix E, and important symbols are listed in Appendix F.

2. Arbitrary Lagrangian–Eulerian theory

While the equations of motion governing arbitrarily curved and deforming fluid films and lipid membranes were pre-sented in Refs. [16,38,42,46], solving the resultant equations is nontrivial. These equations are highly nonlinear and cannot be solved analytically, yet many issues arise when trying to solve them numerically as well. We have described the short-comings of our previous Lagrangian finite element formulation [77,78], which is not appropriate for materials with in-plane flow. Namely, when the surface is discretized and the corresponding mesh travels in-plane with the material, mesh elements become highly distorted and attain nearly singular Jacobians (see Fig. 1a). In such a case, a simple example of vortex flows is not possible. We therefore require a new numerical method to solve the equations of motion.

In this section, we derive an ALE theory for arbitrarily curved and deforming surfaces, which provides equations more amenable to numerical solution. In particular, we seek a description of the material surface which can be easily discretized, with individual elements not undergoing large distortions when material flows in-plane. To this end, we introduce an ALE parametrization of the two-dimensional material, which endows the surface with a mesh that deforms in the normal direction with the material—yet whose in-plane motion can be specified arbitrarily and need not depend on the material flow. We note Ref. [64] introduced a mesh evolving in a similar manner, for the study of scalar fields on evolving surfaces whose time evolution is prescribed. Our ALE description, on the other hand, introduces three new unknowns corresponding to the three components of the mesh velocity. In what follows, we discuss various parametrizations of the surface, and in the ALE case provide the additional three equations required for our problem to be mathematically well-posed. We also describe how the equations governing fluid films and lipid membranes, which are based on an Eulerian surface parametrization, are modified when the ALE parametrization is employed.

2.1. Surface geometry

As discussed in Refs. [42,46], an arbitrarily curved and deforming patch of surface P can be parametrized by either the convected coordinates ξα , which are attached to material points and are convected with the material, or the surface-fixed coordinates θα , which are defined such that a point of fixed θα moves only normal to the surface. Schematics of the movement of points of constant ξα and θα are provided in Figs. 1a and 1b, respectively. Here and from now on, we prescribe Greek indices to span the set {1, 2}, and use the Einstein summation convention in which Greek indices repeated in a subscript and superscript are summed over. As the material occupying a point of constant θα will in general change in time, we formally write θα = θα(ξβ, t) and express the position in terms of either surface-fixed or convected coordinates as

x(θα, t) = x(θα(ξβ, t), t) = x̂(ξβ, t) , (1)where the ‘hat’ accent is used to denote the position expressed in terms of the ξα parametrization (see Fig. 2).


Fig. 2. A schematic of the surface parametrizations. The three different parametric domains, for θα , ξα , and ζα , are shown respectively in green, red, and blue. The mappings from the parametric domains to the patch of surface P are depicted at a single instant in time t with solid, colored arrows. A single material point labeled by the convected coordinates ξβ has corresponding surface-fixed and mesh coordinates θα = θα(ξβ , t) and ζα = ζα(ξβ , t), as shown with the dashed gray arrows. Accordingly, the patch position is equivalently written as (see Eq. (3)) x = x̂(ξα, t) = x(θα, t) = x̌(ζα, t).

As discussed below, the θα parametrization yields a surface description which is in-plane Eulerian and out-of-plane Lagrangian, and is most natural to theoretically model lipid membranes—which are in-plane fluids and out-of-plane elastic solids. Consequently, our theoretical developments [46] used the surface-fixed parametrization throughout. We use the same notation in this work, and review it here. Partial and covariant differentiation with respect to θα are respectively denoted ( ),α and ( );α . The surface-fixed parametrization yields the in-plane basis vectors aα := x,α and unit normal n := (a1 × a2)/|a1 × a2|. The metric and curvature components are respectively given by aαβ := aα · aβ and bαβ := n · x,αβ , with which the mean and Gaussian curvatures are respectively calculated as H := 12 aαβbαβ and K := det(bαβ)/ det(aαβ). A comprehensive geometric description of the surface can be found in Ref. [46] and the references provided therein.

2.2. Surface kinematics

As a point of constant ξα follows a material point over time, the velocity v of a material point is defined as v :=[∂ x̂/∂t]|ξβ , which upon substitution of Eq. (1) and application of the chain rule yields

v = ∂θα

∂t

∣∣∣ξβ

∂x

∂θα+ ∂x

∂t

∣∣∣θα

= ∂θα

∂t

∣∣∣ξβ

aα + ∂x∂t

∣∣∣θα

. (2)

At this point, the parametrization θα is defined such that the second term on the right-hand side of Eq. (2) lies entirely in the normal direction. In this case the normal component of the velocity, v = v · n, satisfies vn = [∂x/∂t]|θα —and a point of constant θα is unaffected by in-plane flow, as shown in Fig. 1b. We accordingly refer to θα as a surface-fixed parametriza-tion. The in-plane contravariant velocity components vα are defined as vα := [∂θα/∂t]|ξβ , such that the velocity v can be written as v = vαaα + vn—indicating our definitions of the normal velocity v and contravariant velocity components vαare consistent with the geometric description of the surface.

The two different parametrizations introduced thus far offer perspectives analogous to the familiar Lagrangian and Eu-lerian formulations from standard continuum mechanics. A point of constant ξα is a material point, so the convected coordinates provide a Lagrangian perspective. A point of constant θα , on the other hand, is independent of the in-plane surface flow and so the surface-fixed coordinates provide an in-plane Eulerian perspective. The material time derivative is calculated as d( )/dt := [∂( )/∂t]|ξβ in the Lagrangian perspective, and as d/dt( ) = vα( ),α + ( ),t for scalar quantities in the in-plane Eulerian perspective, where the partial time derivative is defined as ( ),t := [∂( )/∂t]|θα . We often denote the material time derivative of a quantity with a dot over that quantity, as in v = ẋ or vα = θ̇α . By applying the material time derivative to the basis vectors aα and n, we find ȧα = v ,α and ṅ = −(aα ⊗ n) v ,α , where ⊗ denotes the dyadic or outer product. We calculate the material time derivatives of the metric and curvature components as ȧαβ = v ,α · aβ + v,β · aαand ḃαβ = v;αβ · n. A more detailed description of the surface geometry and kinematics is provided in our past theoretical work [46].


2.3. ALE description and geometry

Thus far, the convected coordinates ξα and surface-fixed coordinates θα were introduced to describe a surface with in-plane fluidity. Lagrangian numerical methods based on convected coordinates are conceptually simpler to develop, yet they cannot capture in-plane flows—which highly distort mesh elements, as discussed previously. Surface-fixed coordinates are most natural for a theoretical description of arbitrarily curved and deforming surfaces with in-plane fluidity, and their numerical implementation requires nodes to only move orthogonally to the surface (see Fig. 1b). A numerical method based on the θα parametrization can model arbitrarily large in-plane flows, yet in rare instances the discretized nodes move to-wards one another as the surface deforms, also shown in Fig. 1b. In the spirit of formulating a completely general numerical method to describe evolving surfaces with in-plane fluidity, we begin by developing an ALE surface parametrization, denoted by the ALE coordinate ζα , which endows the surface with a mesh. The main idea is to define ζα such that the correspond-ing mesh deforms out-of-plane with the material, while its in-plane motion can be specified arbitrarily (see Fig. 1c). We now describe how the geometric description introduced in Sec. 2.1 is modified when expressing quantities in terms of the ALE parametrization ζα .

In general, the material at any ALE coordinate ζα will change over time, as there is in-plane flow relative to the mesh, and the mapping from convected to ALE coordinates is expressed as ζα = ζα(ξβ, t). The surface position can be described equivalently with convected coordinates, surface-fixed coordinates, or the newly introduced ALE coordinates. To this end, any surface position x can be written as

x = x̂(ξα, t) = x(θα, t) = x̌(ζα, t) , (3)where in the last equality and from now on a ‘check’ accent is used to denote the position expressed in terms of the ζαparametrization (see Fig. 2).

Under a change of surface parametrization from θα to ζα , the latter of which can be specified arbitrarily, our geometric description of the surface is modified—as is any quantity with a Greek index. However, quantities transform in such a way that any variable without an index is invariant to the change in parametrization. Thus, for example, v and v are invariant quantities while vα and aαβ are not. As we will see, the continuity equation and vector form of the equations of motion contain no free indices, and can be expressed in terms of any surface parametrization—which shows the utility of our differential geometric developments and notation.

In this work, quantities are expressed in terms of the ζα parametrization by placing a ‘check’ accent over every Greek index, where checked and unchecked indices take the same value. For example, aα̌ := ∂ x̌/∂ζα = x̌,α̌ are the new in-plane basis vectors, aα̌β̌ := aα̌ · aβ̌ are the new metric components, and bα̌β̌ := n · x̌,α̌β̌ are the new curvature components. In this manner, the ALE parametrization is used throughout the rest of this manuscript. We used a similar technique in our previous Lagrangian surface description [78], where we transitioned from the θα to the ξα parametrization. As all quantities in Ref. [78] were written in the ξα representation, all Greek indices should be interpreted as having a ‘hat’ accent to be consistent with our notation.

2.4. ALE kinematics

While the material velocity v is an invariant quantity, it is expressed differently for different surface parametrizations. For example, v := [∂ x̂/∂t]|ξβ for convected coordinates and v = vαaα + vn for surface-fixed coordinates (see Sec. 2.2). In this section, we characterize the kinematics of an arbitrarily curved and deforming surface when the surface is parametrized by the ALE coordinates ζα . Our developments mirror those of Sec. 2.2.

Using the mapping ζα = ζα(ξβ, t), any surface position can be written asx̌(ζα, t) = x̌(ζα(ξβ, t), t) = x̂(ξβ, t) , (4)

which is analogous to Eq. (1). The velocity of a point, v = [∂ x̂/∂t]|ξβ , can be expressed in the ζα representation as

v = ∂ζα

∂t

∣∣∣ξβ

∂ x̌

∂ζα+ ∂ x̌

∂t

∣∣∣ζα

= ∂ζα

∂t

∣∣∣ξβ

aα̌ + ∂ x̌∂t

∣∣∣ζα

, (5)

where in the second equality we substituted aα̌ = x,α̌ . The last term in Eq. (5) describes how the position of a mesh point changes in time, which we denote the mesh velocity vm:

vm := ∂ x̌∂t

∣∣∣ζα

= x′ . (6)

In Eq. (6), the notation ( )′ := [∂( )/∂t]|ζα indicates how a quantity at a mesh point evolves in time. The partial time derivative ( )′ in the ALE parametrization is analogous to the partial time derivative ( ),t in the surface-fixed parametriza-tion, as both describe how a quantity changes at a fixed value of the appropriate coordinates. Note the mesh velocity v mneed not be orthogonal to the surface, in contrast to its surface-fixed analog vn = [∂x/∂t]|θα (see Sec. 2.2). Defining


cα̌ := ∂ζα

∂t

∣∣∣ξβ

= ζ̇ α and c := cα̌aα̌ , (7)

and using Eq. (6), we express Eq. (5) as

v = c + vm . (8)Equation (8) indicates that c is the relative velocity between the material and the mesh. Furthermore, since c · n = cα̌aα̌ ·n and aα̌ · n = 0, Eq. (7)2 shows that the relative velocity lies entirely in the tangent plane to the surface, as shown schematically in Fig. 1c.

The material time derivative of any invariant quantity can be expressed in the ζα representation as

d

dt( ) = ( )′ + cα̌ ( ),α̌ = ( )′ +

(v − vm) · aα̌ ( ),α̌ , (9)

where the relation cα̌ = (v − vm) · aα̌ , obtained from Eqs. (7)2 and (8), is used in the second equality. The acceleration of a point x is calculated using Eq. (9) as

v̇ = v ′ + cα̌ v ,α̌ = v ′ +(

v ,α̌ ⊗ aα̌) (

v − vm) . (10)Finally, in our simulations, the mesh velocity vm is treated as a fundamental unknown. The mesh position is calculated by integrating the mesh velocity over time, formally written as

x̌(ζα, t) = x̌(ζα, t0) +t∫

t0

vm(ζα, t′) dt′ , (11)

where x̌(ζα, t0) is the initial mesh position at time t0.

2.5. Mesh velocity equations

With the introduction of three new unknowns, namely the three components of the mesh velocity vm, three additional governing equations are required for the problem to be mathematically well-posed. One equation is found by taking the dot product of Eq. (8) with the normal vector n and recognizing c · n = 0, yielding

vm · n = v · n , (12)which ensures the mesh and the surface always overlap with one another as the surface deforms. In this sense, all schemes considered are out-of-plane Lagrangian.

The remaining two equations required to close the problem come from specifying the relative velocity c , or equivalently specifying the relationship between vm · aα̌ and v · aα̌ . There are no restrictions on how vm · aα̌ and v · aα̌ are related, so one can specify their relationship arbitrarily. If we were to choose vm · aα̌ = v · aα̌ , for example, we implicitly set ζα = ξαand therefore recover a Lagrangian scheme in which the mesh velocity and material velocity coincide. If, on the other hand, we choose vm ·aα̌ = 0, we implicitly set ζα = θα and recover an in-plane Eulerian scheme in which the mesh moves only in the direction normal to the surface. The theoretical developments of this section allow us to specify v m · aα̌ , or equivalently ζα , arbitrarily as is best-suited to solve the problem at hand (see Fig. 1c). This flexibility is analogous to that of a Cartesian ALE formulation [51], and for this reason we name our scheme ‘arbitrary Lagrangian–Eulerian.’

For the majority of this manuscript we consider an out-of-plane Lagrangian, in-plane Eulerian scheme, for which the mesh velocity satisfies

vm · aα̌ = 0 , (13)which from now on is called a Lagrangian–Eulerian (LE) scheme. As shown in Sec. 6, the LE implementation can be easily modified to produce a pure Lagrangian scheme. The analysis of more general mesh velocity descriptions is left to a future study, as care must be taken to avoid well-known ALE issues arising in the discretized equations—such as the violation of the geometric conservation law [81,82]. In the LE case, such issues do not arise, and one can condense the three constraints on the mesh velocity into a single vector equation, given by

vm = (n ⊗ n) v . (14)Equation (14) provides the three equations necessary to resolve the LE mesh motion, and concludes our theoretical ALE surface description.


3. Fluid film equations

The equations governing an arbitrarily curved and deforming fluid film can be obtained in the form presented below by starting with the lipid membrane equations of Ref. [46], obtained within the framework of irreversible thermodynamics, and setting the bending moduli to zero. Here and from now on, all equations are written in terms of the ALE parametrization by placing ‘check’ accents over all Greek indices, as described in the previous section.

3.1. Strong formulation

The continuity equation for an arbitrarily curved, incompressible two-dimensional material is given by

aα̌ · v ,α̌ = v α̌;α̌ − 2 v H = 0 . (15)Equation (15) is also called the incompressibility constraint and is enforced by the Lagrange multiplier λ = λ(ζα, t)—which is physically the surface tension of the material. The local form of the linear momentum balance is given by

ρ v̇ = ρb + T α̌;α̌ , (16)where ρ is the density, v̇ is the acceleration, b are the body forces, and T α̌ are the stress vectors, namely, the boundary tractions across curves of constant ζα . For fluid films, we find T α̌ = σ α̌β̌a

β̌, where σ α̌β̌ are the in-plane stress components

given by (see Appendix A.1 and Ref. [46] for more details)

σ α̌β̌ = λaα̌β̌ + πα̌β̌ . (17)In Eq. (17), λ is the surface tension enforcing areal incompressibility (15) and πα̌β̌ are the in-plane viscous stresses. For an isotropic and incompressible fluid film, the viscous stresses are found to be

πα̌β̌ = ζ ȧμν aαμ aβν = ζ v,γ̌ ·(aα̌ aγ̌ β̌ + aβ̌ aγ̌ α̌) , (18)

where ζ is the two-dimensional shear viscosity with units of force·time/length or equivalently mass/time.To write the equations of motion in component form, we decompose the body force b in the {aα̌ , n} basis as b =

bα̌ aα̌ + p n, where bα̌ are the in-plane covariant components and p is the pressure drop across the surface. The in-plane

equations of motion are given by

ρ v̇ · aα̌ = ρbα̌ + λ,α̌ + ζ(

aβ̌γ̌ v α̌;β̌γ̌ + K v α̌ + 2 v,α̌ H − 2 v,β̌ bβ̌α̌ − 2 v H,α̌)

, (19)

as shown in Appendix A.2. The left-hand side of Eq. (19) contains the inertial terms, while the right-hand side consists of the body forces, surface tension gradient, and divergence of the viscous stresses, respectively. The viscous forces clearly show the coupling between surface curvature and the in-plane and out-of-plane velocity components. Note Eq. (19) was obtained via other techniques in Refs. [83–86], in the context of stationary and time-evolving fluid surfaces.

The out-of-plane equation of motion, also called the shape equation, is found to be (see Appendix A.2)

ρ v̇ · n = p + 2λ H + ζ(

2 bα̌β̌ v α̌;β̌ − 4 v(2H2 − K )) . (20)

Equation (20) is an extension of the Young–Laplace equation to fluid films with nonzero velocity. Indeed, by setting v = 0, Eq. (20) simplifies to the familiar Young–Laplace equation, p + 2λH = 0. The presence of the in-plane fluid viscosity ζ and in-plane velocity components vα̌ in Eq. (20) leads to nontrivial coupling between the in-plane and out-of-plane equations when the surface is curved, i.e. when bα̌β̌ �= 0.

In our ALE formulation, the surface shape is evolved with the mesh velocity, rather than the material velocity, according to Eq. (11). As such, there are seven unknowns to solve for: three components of the material velocity v , three components of the mesh velocity vm, and the surface tension λ. The corresponding equations are the three components of the equation of motion (19) and (20), the three components of the mesh equation (14), and the incompressibility constraint (15). These seven equations constitute the strong formulation of the problem.

3.2. Boundary conditions

The decomposition of the equations of motion into in-plane and out-of-plane components allows us to determine possi-ble boundary conditions. The first term in parenthesis on the right-hand side of Eq. (19), aβ̌γ̌ vα̌;β̌γ̌ , contains two derivatives of the in-plane velocity components vα̌ . As no higher derivative of vα̌ appears, we accordingly specify either Dirichlet or Neumann boundary conditions at every point on the patch boundary. In analogy to the boundary conditions of a bulk fluid in three dimensions, Dirichlet boundary conditions specify the in-plane velocity component vα̌ , while Neumann boundary conditions specify the in-plane boundary tractions T = σ α̌β̌ να̌ a ˇ , where να̌ = ν · aα̌ and ν is the in-plane unit normal

β


to the surface at its boundary [46]. Accordingly, the patch boundary ∂P is separated into a Dirichlet portion ∂vP and a Neumann portion ∂ tP , such that ∂vP ∩ ∂ tP = ∅ and ∂vP ∪ ∂ tP = ∂P . In this manuscript, only traction-free boundary conditions are considered, for which T = 0 on ∂ tP . General traction boundary conditions will be considered in Part II.

We next consider the shape equation (20), which describes the out-of-plane behavior of the fluid film and also provides an evolution equation for the position x through the normal velocity v . The shape equation contains two spatial derivatives of the position through the curvature components bα̌β̌ . Consequently, at each point on the boundary we specify either the normal velocity or its in-plane gradient along the ν direction, perpendicular to the surface boundary. These boundary conditions are independent of the in-plane boundary conditions, and in this manuscript we always specify the normal velocity of the patch boundary.

Our boundary conditions on the equations of motion are succinctly written as

v = v̄ on ∂vP and T = 0 , v = v̄ on ∂ tP , (21)where v̄ is a known velocity on the boundary and v̄ is a known normal component of the velocity on the boundary. As the equations governing the mesh velocity (14) are algebraic equations which do not contain any derivatives, we do not specify any boundary conditions for the mesh velocity vm.

4. Finite element formulation for fluid films

In this section, we determine the weak formulation of the governing equations presented in Sec. 3.1, subject to the boundary conditions in Sec. 3.2. The weak form is modified to remove numerical inf–sup instabilities arising from the incompressibility of the fluid film, with a method inspired by Dohrmann and Bochev [79]. We provide the function spaces in which the solution to the weak formulation resides, and then use the standard tools of finite element analysis to find an approximate numerical solution to the weak form of the governing equations.

While inertial terms are included in the strong and weak formulations for completeness, they are not included in our simulations of arbitrarily curved and deforming fluid films—as they are negligible in many physical problems of interest. Despite the absence of inertia, the equations of motion are nonlinear due to the many terms involving the surface geometry. Furthermore, time derivatives still remain in the problem because the rate of change of the surface position is contained in the mesh velocity. The only simulation including inertial terms is the lid-driven cavity benchmark problem, in which the mesh is constrained to be fixed and inertia is included purely to demonstrate the validity of our numerical method.

4.1. Weak formulation

Here we derive the weak formulation of the strong form equations provided in Sec. 3.1, in similar fashion to Refs. [78,87]. Let V be the space of functions for the material velocity v and mesh velocity vm, and let � be the space of functions for the surface tension λ. We consider the arbitrary variations δv ∈ V0, δvm ∈ V , and δλ ∈ �, where all elements of V0 ⊂ Vvanish on the Dirichlet portion of the boundary. The variations are contracted with the appropriate strong form equations and integrated over the fluid surface to yield the weak formulation of the problem.

In our previous work [46], we presented surface integrals as being of the general form ∫P(. . .) da, where da is a

differential areal element of the patch P . While such a description is theoretically sound, it is not amenable to numerical integration. We define � := {(ζ 1, ζ 2)} to be the space of all ALE coordinates ζα , shown in blue in Fig. 2, and map areal integrals to � according to∫

P

(. . .)

da =∫�

(. . .)

J m d� . (22)

In Eq. (22), d� := dζ 1 dζ 2 is a differential element of � and J m := √det aα̌β̌ is the Jacobian determinant of the mapping x̌(ζα) : � → P . In a similar way, integrals over the patch boundary ∂P are mapped to integrals over the parametric domain boundary � := ∂� according to∫

∂P

(. . .)

ds =∫�

(. . .)

J m� d� , (23)

where ds is a differential line element of the patch boundary ∂P , d� is a differential line element on the parametric domain boundary, and J m� is the Jacobian determinant of the mapping x̌b(ζ

α) : � → ∂P—in which x̌b refers to the position of the patch boundary. The Dirichlet and Neumann portions of �, denoted �v and �t, map to the patch boundaries ∂vP and ∂ tP , respectively.

To obtain the weak formulation, we begin by contracting the equations of motion (16) with an arbitrary velocity variation δv ∈ V0 and integrating over the patch P to obtain


∫�

δv · ρ v̇ J m d� −∫�

δv · ρb J m d� −∫�

δv · T α̌;α̌ J m d� = 0 ∀ δv ∈ V0 , (24)

where all integrals are mapped to � through Eq. (22). Note that in our previous Lagrangian formulation [77,78], we solved for the material position x and the weak form contained an arbitrary position variation δx. In this case, however, the in-plane fluidity necessitates the velocity v to be the fundamental unknown, such that the weak form is calculated with an arbitrary velocity variation δv . Applying the surface divergence theorem to the third term on the left-hand side in Eq. (24)yields

−∫�

δv · T α̌;α̌ J m d� =∫�

δv ,α̌ · T α̌ J m d� −∫�

δv · T J m� d� , (25)

and the boundary integral is simplified by recognizing the velocity variation δv = 0 on �v by construction and T = 0 on �t(21)2, such that

−∫�

δv · T α̌;α̌ J m d� =∫�

δv ,α̌ · T α̌ J m d� . (26)

Substituting Eq. (26) into Eq. (24) and recognizing T α̌ = σ α̌β̌aβ̌

, with σ α̌β̌ given by Eq. (17), we obtain

Gv :=∫�

δv · ρ v̇ J m d� +∫�

δv ,α̌ · πα̌β̌ aβ̌ J m d�

+∫�

δv,α̌ · aα̌ λ J m d� −∫�

δv · ρb J m d� = 0 ∀ δv ∈ V0 .(27)

The weak form of the mesh equation is found by contracting Eq. (14) with an arbitrary mesh velocity variation δvm, multiplying by a constant αm, and integrating over the surface, which yields

Gm := αm∫�

δvm ·[

vm − (n ⊗ n)v] J m d� = 0 ∀ δvm ∈ V . (28)The factor αm is introduced such that Gm has units of power, in agreement with Gv (27). In our simulations, αm is set to unity. While Eq. (28) corresponds to the LE mesh equation (14), a different mesh equation can be used in our ALE formulation by setting Gm to be the corresponding weak form expression.

Finally, as λ is the Lagrange multiplier corresponding to the incompressibility constraint (15), we multiply Eq. (15) by an arbitrary variation δλ ∈ � and integrate over the patch to obtain

Gλ :=∫�

δλ(aα̌ · v ,α̌

)J m d� = 0 ∀ δλ ∈ � . (29)

Eqs. (27)–(29) are the weak forms corresponding to the strong forms given respectively by Eqs. (16), (14), and (15). By summing them together and introducing a shorthand for the vector of unknowns, u, as uT := (vT, (vm)T, λ)T, its variation δu as δuT := (δvT, (δvm)T, δλ)T, and the space of arbitrary variations U0 as U0 := V0 × V × �, we obtain the overall weak formulation, given by∫

�

δv · ρ v̇ J m d� +∫�

δv ,α̌ · πα̌β̌ aβ̌ J m d� +∫�

δv ,α̌ · aα̌ λ J m d� −∫�

δv · ρb J m d�

+ αm∫�

δvm ·[

vm − (n ⊗ n)v] J m d� + ∫�

δλ(aα̌ · v,α̌

)J m d�

= 0 ∀ δu ∈ U0 . (30)Note the weak form is nonlinear due to the out-of-plane deformations of the fluid film, as well as the inertial terms. Introducing G as the direct Galerkin expression [88] corresponding to the left-hand side of Eq. (30), the weak form can be compactly written as

G(u(ζα, t), δu(ζα)

) := Gv + Gm + Gλ = 0 ∀ δu ∈ U0 . (31)


4.2. Solution spaces

With the weak formulation (30), in what follows, we define the infinite-dimensional spaces in which the surface tension λ, velocity v , and mesh velocity vm reside.

Surface tension solution space The surface tension λ enters the weak form (30) without any gradients, and so we require λonly to be square-integrable. We define the space of all possible surface tension fields, �, as the space of square-integrable functions on the parametric domain �, denoted L2(�). Thus, � is given by

� = L2(�) :={

u(ζα): � →R such that(∫

�u2 d�

)1/2< ∞

}. (32)

Velocity solution space The weak formulation (30) contains a gradient of both the velocity variation δv and the material velocity v , the latter of which is contained in the viscous stresses πα̌β̌ (18). The velocity and velocity variation are both elements of the space of functions V , and so elements of V are required to be square-integrable and have square-integrable gradients in order for the weak formulation to remain bounded. Furthermore, as the lipid membrane weak form requires the second derivatives of elements of V to be square-integrable, we define the space of velocities V as

V := (H2(�))3 . (33)Each of the three Cartesian components of the velocity lies in H2(�): the Sobolev space of order two on �, defined as

H2(�) :={

u(ζα): � →R such that u ∈ L2(�) , u,α̌ ∈ L2(�) , u,α̌β̌ ∈ L2(�)}

. (34)

We also define V0 as the space of functions in V which vanish on �v, written as

V0 :={

u(ζα): � →R3 such that u ∈ V , u∣∣�v

= 0}

, (35)

such that δv ∈ V0.

Mesh velocity solution space The weak formulation (30) contains terms involving the gradient of both the velocity variation δv and the mesh velocity vm. While the mesh velocity gradient is not easily recognized in Eq. (30), it is found once the weak form is linearized and discretized (see Eq. (C.25)). As a result, we require vm ∈ V in order for the weak form to remain bounded.

4.3. Finite-dimensional subspaces

We now choose the finite-dimensional subspaces in which we seek vh , vmh , and λh , which are approximations of the true solutions v , vm, and λ, respectively. The approximate surface position xh is chosen to lie in the same subspace as vh , as is standard in isoparametric finite element methods [89]. To this end, we discretize � into ne (number of elements) non-overlapping elements {�1, �2, . . . , �ne}, such that � = ∪ nee=1 �e and � j ∩ �k = ∅ for j �= k. In all cases considered the parametric domain is discretized with a rectangular grid, as required by our choice of basis functions (see Sec. 4.5), such that all elements have the same dimensions—which is denoted h. The partitioning of the parametric domain naturally leads to finite-dimensional subspaces in which functions are polynomials over single elements and have certain continuity requirements across element boundaries.

The finite-dimensional subspace of velocities, Vh ⊂ V , is defined asVh :=

{u(ζα): � →R3 such that u ∈ (C1(�))3 ∩ V , u∣∣

�e∈ (Q2(�e))3 ∀ �e } , (36)

where Cm(�) denotes the space of scalar functions on � with at least m continuous derivatives, and Qn(�e) is the space of bi-polynomial functions of order n on the parametric element �e . Accordingly, Vh is the space of piecewise bi-quadratic functions with continuous first derivatives over the entire domain �. While in the present formulation for fluid films, first derivatives need not be continuous, they are required to be continuous when modeling more complex systems which resist bending, such as lipid membranes [78] and viscoelastic Kirchhoff–Love shells [90]. We define the subspace V0,h ⊂ V0 to be the space of functions in Vh which are also zero on �v, formally written as

V0,h := Vh ∩ V0 . (37)In the finite element analysis of bulk fluids, it is well-known that choosing the Lagrange multiplier space to be of the

same polynomial order as the velocity leads to an unstable matrix equation, an issue resulting from the inf–sup condition, also called the Ladyzhenskaya–Babuška–Brezzi (LBB) condition [91–93]. We refer the reader to Ref. [94] for an excellent analysis of this numerical instability, which is often avoided in practice by choosing the Lagrange multiplier basis functions


to be one polynomial order lower than the velocity basis functions. As we chose our velocities to be piecewise bi-quadratic functions on � (36), the Lagrange multiplier subspace is accordingly chosen to be continuous, piecewise bi-linear functions on �, written as

�h :={

u(ζα): � →R such that u ∈ C0(�) ∩ � , u∣∣�e

∈Q1(�e) ∀ �e}

. (38)

Even with this choice of basis functions, our scheme is LBB-unstable. As a result, we invoke a projection method devised by Dohrmann and Bochev [79], as described below, to further stabilize our numerical method.

4.4. Inf–sup stabilization

In modeling fluids with finite element methods, it is well-known that LBB errors arise when the velocity and surface tension solution spaces are identical. We initially used bi-quadratic velocities and bi-linear surface tensions (32)–(34), which were successfully used in our previous work [77,78]. However, in the present study our numerical scheme exhibited LBB instabilities (see Appendix B.6.1). Further inspection showed our past work may have unknowingly avoided such instabilities by prescribing the surface tension along the entire boundary. In the spirit of developing a completely general finite element formulation, we seek to modify the numerical method presented thus far to remove the LBB instability.

We note there are many techniques to overcome the LBB instability, for example (i) using lower-order shape functions for the Lagrange multipliers [95], (ii) reduced and selective integration of the Lagrange multiplier equations [96,97], (iii) stabilization methods [98], and (iv) macroelement approaches [99,100]. All of these methods are valid in different cases. In this section, we describe how our weak form is modified by a technique developed by Dohrmann and Bochev [79] to remove LBB instabilities given general polynomial spaces for the velocity and surface tension; the analysis of other methods is left to a future study. The main idea underlying the Dohrmann–Bochev method is to locally project the surface tension onto a space of discontinuous, piecewise linear functions, and to energetically penalize the difference between the projected and unprojected surface tensions. The Dohrmann–Bochev method is thus based on two equations: one which projects the surface tension, and another which penalizes surface tension deviations from the projected space in a manner suitable for finite element analysis. A thorough description of the Dohrmann–Bochev method is provided in Ref. [89], and we follow their notation in this manuscript. The details of our numerical implementation can be found in Appendix B.6.

4.4.1. TheoryWe begin by specifying the space of piecewise linear, discontinuous basis functions �̆ onto which the surface tension is

projected, given by

�̆ :={

u(ζα): � →R such that u∣∣�e

∈ P1(�e) ∀ �e}

, (39)

where Pn(�e) is the space of polynomial functions of order n on the parametric element �e . Note that while piecewise bi-linear functions can be continuous on quadrilateral elements (38), piecewise linear functions cannot be. Accordingly, the space �̆ is discontinuous and over a single element �e : �̆

∣∣�e

=P1(�e).We next introduce the projection of the surface tension and its arbitrary variation, denoted λ̆ and δλ̆, respectively, such

that λ̆, δλ̆ ∈ �̆. The L2-projection of the surface tension, λ̆, is defined by∫�

δλ̆(λ − λ̆) d� = 0 ∀ δλ̆ ∈ �̆ . (40)

As δλ̆ belongs to a space of linear functions which are discontinuous across elements, Eq. (40) can be considered separately for individual elements �e , and is equivalently expressed as∫

�e

δλ̆(λ − λ̆) d� = 0 ∀ δλ̆ ∈ P1(�e), ∀ �e . (41)

Deviations between λ̆ and λ are penalized in the weak form by subtracting the term

GDB :=αDB

ζ

∫�

(δλ − δλ̆) (λ − λ̆) d� (42)

from the left-hand side of Eq. (30). In Eq. (42), ζ is the two-dimensional fluid shear viscosity and αDB is a computational parameter having units of J m which, as in Ref. [79], is chosen to be unity. The units of αDB ensure that Eq. (42) is dimen-sionally consistent with the other terms in the weak form (30). The weak formulation is now given by


∫�

δv · ρ v̇ J m d� +∫�

δv ,α̌ · πα̌β̌ aβ̌ J m d� +∫�

δv,α̌ · aα̌ λ J m d� −∫�

δv · ρb J m d�

+ αm∫�

δvm ·[

vm − (n ⊗ n)v] J m d� + ∫�

δλ(aα̌ · v,α̌

)J m d� − α

DB

ζ

∫�

(δλ − δλ̆) (λ − λ̆) d�

= 0 ∀ δu ∈ U0 , (43)and the direct Galerkin expression G in Eq. (31) is redefined such that

G(u(ζα, t), δu(ζα)

) := Gv + Gm + Gλ − GDB = 0 ∀ δu ∈ U0 . (44)We provide the details of our Dohrmann–Bochev implementation in Appendix B.6, within which Appendix B.6.1 demon-strates the success of this method.

4.5. Summary of numerical solution method

At this point, we seek an approximate solution to the weak formulation, as presented in Eqs. (43) and (44). To this end, we discretize the fundamental unknowns and their arbitrary variations. We then obtain the residual equations, discretize them temporally, and solve the resulting equations via Newton–Raphson iteration. Our techniques are briefly summarized here, however, extensive details of our numerical implementation are provided in Appendix B.

We first introduce the basis functions {NI (ζα)} and Lagrange multiplier basis functions {N̄ J (ζα)} such that Vh =(span {NI (ζα)})3, and �h = span {N̄ J (ζα)}. The fundamental unknowns are discretized as

vh(ζα, t) = [N] [v(t)] , vmh (ζα, t) = [N] [vm(t)] , and λh(ζα, t) = [N̄] [λ̄(t)] , (45)

where [N] and [N̄] are shape function matrices and [v(t)], [vm(t)], and [λ̄(t)] are the velocity, mesh velocity, and surface tension degree of freedom vectors, respectively. We introduce the shorthand [u(t)] = ([v(t)]T, [vm(t)]T, [λ̄(t)]T)T as the col-lection of all degrees of freedom in the discretized system. The arbitrary variations δv , δvm, and δλ are discretized with the same basis functions as

δv(ζα) = [N] [δv] , δvm(ζα) = [N] [δvm] , and δλ(ζα) = [N̄] [δλ̄] , (46)according to a Bubnov–Galerkin approximation, with [δv], [δvm], and [δλ̄] collectively gathered into [δu]. The weak formu-lation (44) can then be written as [δu]T[r(t)] = 0 for all [δu], which is satisfied when the residual vector

[r(t)] := ∂G∂[δu] = [0] . (47)

To solve Eq. (47) at a set of N discrete times {t1, t2, . . . , tN}, we assume [u(tn)] is known and solve for [u(tn+1)] := [u(tn)] +[�u(tn+1)] according to the Newton–Raphson method. Again, a detailed description of our numerical procedure is provided in Appendix B.

In our implementation, the spaces Vh (36) and V0,h (37) involve C1-continuous basis functions. We maintain basis func-tion continuity across elements by using uniform B-spline basis functions, which have the advantage of naturally enforcing arbitrary continuity requirements yet the complication of non-interpolatory basis function coefficients, and the requirement of a rectangular parametric mesh, as well as basis functions spreading over multiple elements. The method of using B-spline basis functions within an isoparametric finite element framework is in the spirit of Ref. [101], and detailed in Ref. [102]. We use the algorithms described in Ref. [103] to efficiently calculate the basis functions and their derivatives.

We have now concluded our discussion of the LE finite element formulation, and a high-level overview of our code structure can be found in Algorithm 1 of Appendix C.6.

5. Numerical simulations

We now present several results from our LE finite element formulation to validate the robustness of the method and demonstrate its capabilities. The numerical implementation of the method is tested with problems of increasing complexity, and results are compared to known analytical solutions whenever possible. The first test cases involve fluid flows on flat planes, and once several cases are validated we move on to study fluid flows on fixed, curved surfaces. In the scenarios mentioned thus far, the mesh is constrained to remain stationary and the mesh velocity is not solved for. In our last example, the entire LE implementation is tested by modeling an initially cylindrical fluid film, which is allowed to deform over time. The simulations show that fluid films are unstable with respect to long wavelength perturbations, which may explain why bubbles are often observed and long, cylindrical fluid films are not. Namely, in any experimental system, we expect the latter to break up and form bubbles. A linear stability analysis is performed to calculate the critical length above


which the cylinder becomes unstable, as well as the time scale of the instability. We show that the analytical predictions for the critical length and the time scale of the fastest growing unstable mode agree quantitatively with our simulation results. Moreover, we find both theoretically and computationally that cylindrical fluid films are stable to non-axisymmetric perturbations.

5.1. Fluid flow on a flat plane

We first consider the simplest test case, fluid flowing on a flat plane, for which the surface tension can be equivalently thought of as the negative surface pressure and the governing equations simplify to the two-dimensional incompressible Navier–Stokes equations (see Appendix A.2.1). While the problems considered in this section are easily solved using standard Cartesian finite element methods, we solve them using our nonlinear isoparametric finite element framework, in which differential geometry is used to express the surface position and fundamental unknowns in terms of curvilinear coordinates. Thus, even these simple problems serve as important benchmarks for our LE finite element implementation.

In our simulation of fluid flow on a flat plane, the mesh is constrained to be fixed, such that v m = 0, and we do not solve for the mesh velocity. Furthermore, as the mesh is flat, there is no motion in the out-of-plane direction and we solve only for the x- and y-components of the fluid velocity as well as the surface tension. The three corresponding strong form equations are the continuity equation (15) and the two in-plane Navier–Stokes equations (A.22)3. We simulate five scenarios with increasingly complex solutions: (i) a hydrostatic fluid, with zero velocity and linear surface tension; (ii) Couette flow between parallel plates, with linear velocity and zero surface tension; (iii) Couette flow between parallel plates under the influence of a quadratic body force, with linear velocity and cubic surface tension; (iv) Hagen–Poiseuille flow in a channel, with quadratic velocity and linear surface tension; and finally (v) the lid-driven cavity problem, for which no analytical solution is known. Only the lid-driven cavity result is discussed in the main text, and the validation of the first four cases against analytical solutions is left to Appendix D.

For the numerical results presented in this manuscript, we neglect inertial terms in all cases except the lid-driven cavity, for which inertial terms are evaluated using a backwards Euler temporal discretization. The contribution of inertial terms to the tangent matrix and residual vector, for the limited case of a fixed surface, is provided in Appendix C.3.1. Inertial terms are included in the lid-driven cavity problem only to further validate our numerical implementation with flows at moderate Reynolds numbers.

5.1.1. Lid-driven cavityA schematic of the problem is shown in Fig. 3a. Fluid in a square cavity with stationary walls is driven by a top lid

which moves to the right at constant speed V . We solve for the flow field and surface tension in the cavity, which is taken to be a unit square in the x–y plane. The boundary condition v = 0 is imposed on the sides and bottom of the square domain, and v = V ex is set on the top edge. There is a choice in what velocity to specify at the top two corners of the domain where the stationary edges meet the moving top lid. At these locations we set v = 0, as done in Ref. [104]. Furthermore, as only gradients of the surface tension enter the equations governing a flat, two-dimensional fluid, the surface tension λ is indeterminate up to a constant. Consequently, λ is specified to be zero at the center of the domain, located at (x, y) = (0.5, 0.5).

We first set the inertial terms to zero and solve for the Stokes flow result at Reynolds number Re := ρ V L/ζ = 0, for which the x-velocity is plotted in Fig. 3b. The flow field is moving towards the right at the top of the domain, and the presence of the right wall requires the fluid to be recirculated towards the left side in the bulk of the domain. We then include inertial terms and set Re = 100 by setting ρ = 1 and ζ = 0.01 (we already have V = 1 and L = 1). The corresponding x-velocity is plotted in Fig. 3c. Relative to Fig. 3b, Fig. 3c shows that at the top of the domain, inertia pulls the fluid to the right in the direction of the moving top lid. In Figs. 3d and 3e, the x-velocity and surface tension are plotted at different Reacross a vertical cross-section through the domain. Our results for the lid-driven cavity problem agree with those of other numerical studies [105,106], as plotted in Fig. 3d.

Our final task for the lid-driven cavity problem is to analyze how our simulations converge to the true solution as we increase the number of elements. We consider the case with no inertia, i.e. Re = 0, and calculate the error as a function of the length (and width) of a single element, denoted h. As the exact solution is unknown, we treat the solution on the finest, 256 × 256 mesh as the true solution. We calculate the L2-error for the velocity, denoted ||v − vh||0 and defined as

∣∣∣∣v − vh∣∣∣∣0 :=(∫

�

|v − vh|2 d�)1/2

, (48)

where v is the true velocity and vh is the approximate velocity found on coarser meshes with elements of size h × h. The L2-error is denoted with a subscript ‘0’ because L2(�) = H0(�), i.e. the Sobolev space of order zero. A plot of ||v − vh||0 as a function of 1/h is shown in Fig. 3f, in which we see a linear scaling of the L2-error with the mesh size h. The numerical treatment of the corner nodes may explain why our simulations do not converge faster: at each mesh size, we set v = 0 at the corner nodes and v = V ez at the adjacent nodes on the top edge, however, as the number of elements increases, these two nodes move closer together such that we solve a slightly different problem at each mesh size.


Fig. 3. Lid-driven cavity problem. (a) Schematic of the problem, in which fluid in a cavity (shown in white) of unit height and width is surrounded on three sides by rigid walls. The lid of the cavity is dragged to the right at constant speed V , such that at the top edge the fluid travels towards the right with the same speed. In all numerical calculations, we set V = 1. (b),(c) Plots of the x-velocity throughout the domain, on a 128 × 128 mesh, at Reynolds number Re = 0 (b) and Re = 100 (c). In both cases, fluid flows to the right at the top of the domain and then recirculates to the left in the center. When Re = 100, the fluid is pulled more towards the top right corner of the domain. The outer contour line shows where vx = 0, and the inner contour depicts where vx = −0.12. (d),(e) Plots of the x-velocity (d) and surface tension (e) on a 128 × 128 mesh at the vertical cross section x = 0.5. (d) The x-velocities in our simulations (solid lines) agree with Ref. [105] (red circles) at Re = 0, and Ref. [106] (black squares, blue triangles) at Re = 100 and Re = 400, respectively. (e) The surface tension has small differences for Re = 0, 100, and 400. The point where the three lines meet is the center of the domain, (x, y) = (0.5, 0.5), where we set λ = 0. (f) Plot of the velocity L2-error as a function of the element width h, for Re = 0. As no exact solution is known, we calculate the error relative to the solution on a 256 × 256 mesh. Data is shown in blue; the dashed red line is a reference with slope −1 to show the scaling of the error.

Due to the moving top lid being in contact with the stationary walls at the top two corners of the domain, the lid-driven cavity solution is known to have large surface tension spikes at the corners [104]. For this reason, numerical studies generally use meshes which are very finely discretized at the corners. We find the surface tension spikes vary significantly with the mesh size, for example, on a 16 × 16 mesh λ is O(102) at the corners while on a 128 × 128 mesh it is O(103). These spikes dominate the error calculation, which even on the 128 × 128 mesh is O(1) relative to the solution on a 256 × 256 mesh. The surface tension spikes are plotted in Appendix B.6.1, in which the success of the Dohrmann–Bochev method in removing inf–sup instabilities is also shown. As the surface tension error does not converge in this situation, in Appendix D.3 we show how it converges in the benchmark problem of Couette flow with a body force.

With the numerical results provided thus far, as well as those of Appendix D, we conclude our analysis of fluid flow on a fixed, flat plane. Our general ALE finite element framework, based on a curvilinear coordinate description via the machinery of differential geometry, has successfully reproduced the classical benchmarks of hydrostatic flow, Couette flow, Hagen–Poiseuille flow, and lid-driven cavity flow. We are therefore confident our finite element method can model arbitrary flows on flat surfaces.

5.2. Fluid flow on a stationary, curved surface

After testing our code on the simplest case of fluid flow on a flat plane, we turn to study fluid flows on stationary, curved surfaces. In such problems the mesh is fixed, and thus once again the mesh velocities are not solved for. However, a complication arises because for a fixed surface, the normal component of the material velocity v = v · n = 0. In practice, there are two ways we could handle this problem. First, we could posit that our arbitrary velocity variation δv = δvα̌aα̌ , where aα̌ is known for the fixed surface. However, such a method cannot be easily extended to model a deforming fluid film: when the surface is deforming, the velocity v is represented as v = vα̌aα̌ + vn, and in a fully implicit numerical scheme vα̌ , aα̌ , v , and n are all unknown.

We employ a different approach by representing the velocity and its arbitrary variation in a Cartesian basis, including both the in-plane and shape equations in our weak formulation, and enforcing v = v · n = 0 with a Lagrange multiplier—


which is interpreted physically as the pressure drop required to constrain the fluid to the surface. In this section, we describe how both the strong and weak formulations are modified by the normal pressure p being an unknown Lagrange multiplier field. Our method is then tested by modeling fluid flow on a fixed, bulged cylinder. Numerical results are compared with analytical solutions, and we calculate how our numerical error decreases on mesh refinement.

5.2.1. Strong form modificationIn satisfying the constraint v = 0 with a Lagrange multiplier, we include the shape equation (20) in our description of

the fluid film. In the shape equation (20), the pressure drop p is an unknown Lagrange multiplier field, which at every point on the surface takes the requisite value to satisfy v = 0. There are thus five unknowns: the three components of the velocity, the surface tension, and the normal pressure drop, and five corresponding equations: the continuity equation (15), the two in-plane equations (19), the shape equation (20), and the constraint v = 0.

5.2.2. Weak form modificationWith the pressure drop p being a fundamental unknown, the arbitrary pressure variation δp is expected to enter the

weak formulation. To understand how δp will appear, we take the variation of the virtual work associated with moving the fluid film in the normal direction and obtain

δ

(∫�

v · p n J m d�)

=∫�

δv · p n J m d� +∫�

v · δp n J m d� . (49)

The first term on the right-hand side of Eq. (49) is already contained in Eq. (43) through the body force term. Assuming no in-plane body forces (bα̌ = 0), treating the pressure p as a fundamental unknown, again recognizing inertia is negligible, and removing mesh velocity degrees of freedom yields a modified weak form (cf. Eq. (43))∫

�

δv,α̌ · πα̌β̌ aβ̌ J m d� +∫�

δv ,α̌ · aα̌ λ J m d� +∫�

δλ(aα̌ · v ,α̌

)J m d�

−∫�

δv · p n J m d� −∫�

δp v · n J m d� − αDB

ζ

∫�

(δλ − δλ̆) (λ − λ̆) d�

= 0 ∀ δv ∈ V0, δp ∈ P, δλ ∈ � ,

(50)

where P is the space of pressure solutions and arbitrary pressure variations. The weak form (50) contains no gradients of pressure and thus it is theoretically sound for us to choose P as the space of square-integrable functions on �, i.e. L2(�). However, we simplify our finite element analysis by using the same basis functions for the velocity and pressure. In accordance with Eqs. (33) and (34), we define

P := H2(�) . (51)The structure of Eq. (50) indicates the pressure variation δp enforces the normal constraint v · n = 0, in the same way

the surface tension variation δλ enforces the incompressibility constraint aα̌ · v,α̌ = 0. With the weak formulation (50), an identical procedure to that of Sec. 4 is followed to linearize and discretize the equation, calculate the tangent matrix and residual vector, and then iteratively solve for the unknowns via Newton–Raphson iteration. The approximate pressure ph is an element of the finite-dimensional subspace Ph ⊂P , which is chosen to be

Ph :={

u(ζα): � →R such that u ∈ C1(�) ∩P , u∣∣�e

∈ Q2(�e) ∀ �e}

(52)

in accordance with the finite-dimensional space of velocities Vh (36). The pressure can then be expressed in terms of the same set of basis functions, {NI (ζα)}, used for the velocities. As mentioned previously, our choice of P (51) and Ph(52) is purely for convenience in our numerical implementation. The details of the modifications to our finite element implementation are provided in Appendix C.3.

5.2.3. Flow on a fixed cylinder with a bulgeIn our numerical implementation, we consider fluid flowing on a fixed, bulged cylinder, as shown in Figs. 4a–4c. Our

boundary conditions are shown schematically in Fig. 4a, where constant inflow and outflow of the fluid is prescribed at the entrance and exit of the cylinder, respectively. The surface tension is specified at a single point on the boundary, as only gradients of λ enter the in-plane equations. The bulge in the center leads to nontrivial velocity and surface tension profiles due to the coupling between curvature and fluid flow, and the symmetry of the surface shape allows us to determine the analytical solution (see Appendix A.2.3). The bulged cylinder thus serves as a useful benchmark problem for our numerical method, in the study of flows on curved yet fixed surfaces.


Fig. 4. Fluid flow on a fixed cylinder with a bulge. (a) Schematic of the problem. Fluid enters at the left edge at velocity V ez and exits at the right edge at the same velocity. The surface tension λ is specified to be unity at a single point on the boundary. In the numerical calculation, no assumption about axisymmetry is made. (b) Radius as a function of axial position, with a 4% bulge in the central region. (c) A coarse 10 × 10 mesh of the bulged cylinder. (d–f) Plots of the surface tension (d), z-velocity (e), and normal pressure (f) as a function of axial position. We plot the deviation in these quantities from their analytical value at the left edge of the cylinder (z = 0), with axes scaled for convenience. The solid red lines are the analytical solutions and the dashed blue lines are our numerical results. Calculations were made on a 128 × 128 mesh. (g)–(i) Plots of the L2-error according to Eq. (48), relative to the analytical solution, as we change the number of elements. In (g), we refine in both the z and θ directions, such that there are the same number of elements in each direction. In (h), there are 128 θ -elements and the number of z-elements varies from 8 to 128 in powers of two. In (i), there are 128 z-elements and between 8 and 128 θ -elements, again in powers of two. In (c)–(i), the length L = 4, 000 is chosen for analytical solutions to be sufficiently smooth (see Appendix A.2.3).

The position of the bulged, cylindrical surface is given by

x(θ, z) = r(z) er(θ) + z ez , (53)where θ and z are the polar angle and axial distance, respectively, of a standard cylindrical coordinate system. The radius r(z) is independent of θ because the surface is axisymmetric. Our choice of radial profile is given in Appendix A.2.3 and shown in Fig. 4b. We define hθ and hz as the width of an element �e in the θ and z directions, respectively, and denote a mesh with 16 elements in the θ -direction and 32 elements in the z-direction, for example, as a 16 × 32 mesh. A coarse 10 × 10 mesh of the bulged cylinder is shown in Fig. 4c. The surface tension, z-velocity, and normal pressure are calculated numerically on a 128 × 128 mesh, and compared to their analytical counterparts in Figs. 4d–4f, respectively. In all cases, our numerical results show excellent agreement with the analytical calculations.

We conclude our analysis of the fixed, bulged cylinder by calculating the L2-error of the velocity, surface tension, and normal pressure. We also calculate the error between our numerical surface position xh and the true position x given in Eq. (53), which is a function of how closely our basis functions can represent the analytical geometry. Figs. 4g–4i show three different plots of the error: changing hθ and hz together (4g), changing hz with fixed hθ (4h), and changing hθ with


fixed hz (4i). Figs. 4g and 4h are nearly identical, indicating θ -refinement has little effect on the errors, an observation which is confirmed with Fig. 4i. As the analytical solution is only a function of z, it is unsurprising that refining in θhas no significant effect on the errors. In particular, θ -refinement only makes observable changes to our numerical surface representation when there are few elements in the θ -direction. In Figs. 4g and 4h, the errors in the position, velocity, and normal pressure converge quadratically on mesh refinement. However, the error in λ is approximately constant. As discussed in Appendix A.2.3, a cylinder length L = 4, 000 is chosen, with radius changes over extended length scales, to avoid discontinuities in the normal pressure. For such cylinders, r′(z) � 1 and r′′(z) � 1, and therefore λ(z) ≈ λ0 = 1 everywhere (see Eq. (A.37))—irrespective of the shape of the bulge. Analytically, the difference between λ and λ0 is O(10−4) and occurs when the radius changes due to nonzero r′(z), r′′(z), and z-velocity (A.37). Numerically, errors in λ are also O(10−4) and occur when the radius changes due to the difference between the numerical and analytical position x; these errors persist along the length of the cylinder (Fig. 4d). As can be seen from Fig. 4h, the errors in position x are � 10−4 for all simulated meshes, and therefore the error in λ remains approximately constant at 10−4. We expect the λ error to decrease as one further refines the mesh, thus reducing errors in the position. However, these simulations are prohibitively expensive in our current implementation and require parallelization of our code. Our error calculations, captured in Figs. 4g–4i, conclude our analysis of fluid flows on fixed, curved surfaces.

5.3. Curved and deforming fluid films

In this section, we demonstrate the capabilities of our full LE finite element implementation by studying the stability of cylindrical fluid films which have a constant pressure drop across their surface. It is already known that given a surface tension and pressure drop, spherical fluid films of radius R satisfy the Young–Laplace equation p = 2λ/R and are stable [107]. The cylindrical fluid films under consideration, however, are found to be unstable to axisymmetric perturbations; the large, nontrivial shape changes resulting from the instability are studied here. Our numerical results are compared with analytical results from a linear stability analysis, and the two are found to be in excellent agreement. Moreover, we find the fluid film instability is mediated by the in-plane flow resulting from the initial shape perturbation, thus demonstrating the importance of our ALE framework in studying two-dimensional surfaces with in-plane flow. Our general LE implementation is also employed to show cylinders of any length are stable to non-axisymmetric perturbations, as confirmed analytically. We end by showing our results are independent of our choice of time step and mesh size, thus demonstrating the necessary convergence of numerical solutions.

5.3.1. Stability of axisymmetric perturbationsTo begin, the position of an unperturbed cylinder of radius R and length L is given by

x(θ, z) = R er(θ) + z ez , (54)with the polar angle θ ∈ [0, 2π) and the axial length z ∈ [0, L]. For the boundary conditions v = 0 on both edges of the cylinder, a valid base state solution is given by v = 0 and λ = λ0 everywhere, as shown in Appendix A.3. In this case the shape equation (20) simplifies to the Young–Laplace equation λ0 = p R , with p being the constant pressure drop imposed across the fluid surface.

At time t = 0, we perturb the radius of the stationary cylinder such that the initial position is given by

x(θ, z; t = 0) = R[

1 + � sin(2π z

L

)]er + z ez , (55)

where the small dimensionless parameter � is set to 0.01 in our simulations. A schematic of the initial surface shape is shown in Fig. 5a. We evolve the fluid film from its initial state using our LE method and observe if it is stable or un-stable with respect to the initial perturbation. Over the course of our simulation, we maintain a constant pressure drop across the fluid surface. This pressure drop enters as a body force ρb = pn, and its contribution to the tangent ma-trix and residual vector is provided in Appendix C.4. We find the cylinder is stable to the initial perturbation when its length L < 2π R and unstable when L > 2π R . In the latter case, the initial perturbation continues to grow and eventually reaches a configuration that has spherical bulbs on the two ends (Figs. 5b–5f, supplemental movie E.1, video provided at youtu.be/FUx8fGXuzqY). These spherical shapes are believed to form because they are the stable surfaces compatible with our boundary conditions and incompressibility constraint. We proceed to use simple physical arguments to understand why the initial perturbation is unstable.

5.3.2. Physical explanation of the instabilityThe instability arises because our initial shape perturbation changes the mean curvature of the surface, which in turn

changes the surface tension through the Young–Laplace equation. The resultant surface tension gradients then drive in-plane fluid flow, as can be seen from the in-plane equations (19). When L > 2π R , fluid flows from the narrow region of the cylinder to the wide region (see Fig. 5a), resulting in an unstable film. However, when L < 2π R , the surface flow is directed from the wide region to the narrow region, which causes the initial bulge to shrink over time such that the surface returns to its cylindrical configuration.

https://youtu.be/FUx8fGXuzqY


Fig. 5. Curved and deforming fluid film. (a) Schematic of the initial film shape, according to Eq. (55), where the maximum radius perturbation is 1%. (b)–(f) Snapshots of an unstable and deforming film at t = 15 (b), t = 20 (c), t = 25 (d), t = 30 (e), and t = 35 (f), where the color indicates the surface tension. At early times (b), the shape perturbation results in surface tension gradients which drive a destabilizing flow. At late times (f), spherical bulbs appear on the ends of the cylinder. At all times, we enforce v = 0 on both edges. A video of the instability can be found at youtu.be/FUx8fGXuzqY or supplemental movie E.1. Simulation parameters are R = 1, L = 10, λ0 = 1, ζ = 1, �t = 0.1, hθ = 1/10, and hz = 1/40.

To understand this general idea in more detail, we begin with the Young–Laplace equation, written as λ = −p/(2H). For an unperturbed cylinder, H = −1/(2R). The initial perturbation (55) alters the mean curvature of the film by modifying both radii of curvature: it changes the radius of a circular cross-section of the cylinder, and the sinusoidal shape along zintroduces a nonzero radius of curvature in the axial direction. Analytically, the mean curvature H of the initially perturbed shape (55) is calculated according to

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